Possible fixes (that don’t require throwing out data )
Clustered Standard Errors: using either the bootstrap or a sandwich estimator to adjust the standard errors up
Fixed effects: make a dummy for each group and include it in the model, (potentially) eliminating the autocorrelation while also controlling for unobserved differences across groups
Fixed Effects Models give every group its own y-intercept. The effects of “Texas” vs. “Tennessee” are assumed to be fixed and unrelated.
In this sense, you’re usually assuming a fixed effect any time you run a standard OLS model.
Random Effects Models assume that group intercepts are random draws from a common distribution. They can vary just like any other random variable, but they vary in a predictable way
An example: students in classrooms
You want to estimate the effect of time spent studying on student achievment
But you also want to account for differences between teachers: you suspect some are slightly better than others.
flowchart LR
Z[School] --> A
Z --> G
Z --> L
A[Classroom 1] --> B(Student A)
A --> C(Student B)
A --> E(Student D)
G[Classroom 2] --> H(Student H)
G --> I(Student I)
G --> K(Student K)
L[Classroom 3] --> M(Student M)
An example: students in classrooms
You calculate some averages and your results look like this:
Instructor
Number of students
Average SAT score
Teacher 1
50
1029
Teacher 2
25
1020
Teacher 3
5
1120
Under a fixed effects assumption, each teacher’s effect is a fixed parameter, and knowing the mean score for teacher 1 and 2 doesn’t tell you anything about what the mean score should be for teacher 3.
Under a random effects assumption, the teacher effect is drawn from a larger distribution that has a central tendency and a variance. So you might look at the value for teacher 3 and conclude: “this seems implausible, in reality, this effect should probably be closer to the effect for the other two”
Lack of independence poses similar problems to what we observed with clustered samples: we will generally overstate our level of confidence in estimates. We actually have very few truly independent observations here.
Failure to account for cycles can also just lead to poor prediction in the face of complex trends. The overall trend in total airline flights looks like this:
Taking the difference of both polyarchy and GDP in this model should eliminate the time trend effect. But it also alters what we’re actually regressing.
Pro: allows a much more complex model specification and also tests for autocorrelation
Con: interpretation is weird, and it adds a lot of parameters and inevitable multicollinearity
Other fixes
Including a fixed effect for time can work if you have multiple observations for different periods or if you only suspect a cyclical trend
More complicated autocorrelation structures can be modeled with a regression, and then OLS can be used on the residuals from that model
ACF plots
ACF plots can be used to visualize autocorrelations. If the lagged correlation is above the blue line, then there’s statistically meaningful autocorrelation.
